en

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular >

integral of sqrt(12+8x-3x^2)

Solution

\int 12 + 8 x - 3 x^{2} d x

Solution

\frac{13}{3 3} (2 arcsin (\frac{3}{2 13} x - \frac{2}{13}) + sin (2 arcsin (\frac{3}{2 13} x - \frac{2}{13}))) + C

Solution steps

\int 12 + 8 x - 3 x^{2} d x

Complete the square

12 + 8 x - 3 x^{2} : - 3 (x - \frac{4}{3})^{2} + \frac{52}{3}

= \int - 3 (x - \frac{4}{3})^{2} + \frac{52}{3} d x

Apply u-substitution

: \int \frac{- 9 u ^{2} + 52}{3} d u

= \int \frac{- 9 u ^{2} + 52}{3} d u

\frac{- 9 u ^{2} + 52}{3} = \frac{1}{3} - 9 u^{2} + 52

= \int \frac{1}{3} - 9 u^{2} + 52 d u

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= \frac{1}{3} \cdot \int - 9 u^{2} + 52 d u

Apply Trigonometric Substitution

: \int \frac{52}{3} cos^{2} (v) d v

= \frac{1}{3} \cdot \int \frac{52}{3} cos^{2} (v) d v

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= \frac{1}{3} \frac{52}{3} \cdot \int cos^{2} (v) d v

Rewrite using trig identities

= \frac{1}{3} \frac{52}{3} \cdot \int \frac{1 + cos ( 2 v )}{2} d v

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= \frac{1}{3} \frac{52}{3} \cdot \frac{1}{2} \cdot \int 1 + cos (2 v) d v

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= \frac{1}{3} \frac{52}{3} \cdot \frac{1}{2} (\int 1 d v + \int cos (2 v) d v)

\int 1 d v = v

\int cos (2 v) d v = \frac{1}{2} sin (2 v)

= \frac{1}{3} \frac{52}{3} \cdot \frac{1}{2} (v + \frac{1}{2} sin (2 v))

Substitute back

= \frac{1}{3} \frac{52}{3} \cdot \frac{1}{2} (arcsin (\frac{3}{2 13} (x - \frac{4}{3})) + \frac{1}{2} sin (2 arcsin (\frac{3}{2 13} (x - \frac{4}{3}))))

Simplify

\frac{1}{3} \frac{52}{3} \cdot \frac{1}{2} (arcsin (\frac{3}{2 13} (x - \frac{4}{3})) + \frac{1}{2} sin (2 arcsin (\frac{3}{2 13} (x - \frac{4}{3})))) : \frac{13}{3 3} (2 arcsin (\frac{3}{2 13} x - \frac{2}{13}) + sin (2 arcsin (\frac{3}{2 13} x - \frac{2}{13})))

= \frac{13}{3 3} (2 arcsin (\frac{3}{2 13} x - \frac{2}{13}) + sin (2 arcsin (\frac{3}{2 13} x - \frac{2}{13})))

Add a constant to the solution

= \frac{13}{3 3} (2 arcsin (\frac{3}{2 13} x - \frac{2}{13}) + sin (2 arcsin (\frac{3}{2 13} x - \frac{2}{13}))) + C

Popular Examples

integral of (3t^2-sqrt(5t)+2)

\int (3 t^{2} - 5 t + 2) d t

integral of 1/(6+e^{-8x)}

\int \frac{1}{6 + e ^{- 8 x}} d x

integral of (sin(2x)+cos(x))^2

\int (sin (2 x) + cos (x))^{2} d x

integral of 5/(sqrt(x^2+4))

\int \frac{5}{x ^{2} + 4} d x

integral of 3x^2-7x+2

\int 3 x^{2} - 7 x + 2 d x